Let \((F,+,\cdot)\) and \((F^\prime,\oplus,\odot)\) be two fields. The map \(f:F\mapsto F^\prime\) is called a field homomorphism, if for all \(a,b\in F\): $$\begin{array}{rcl}f(a+b)&=&f(a)\oplus f(b),\\f(a\cdot b)&=&f(a)\odot f(b),\\f(1_{F})&=&f(1_{F^\prime}),\end{array}$$ where $1_{F}$ and $1_{F^\prime}$ denote the respective multiplicative neutral elements of the respective two fields.
The complex conjugation $f:\mathbb C\to\mathbb C$ is a field homomorphism (even an endomorphism), since $(\mathbb C,+,\cdot)$ is a field and $$\begin{array}{rcl} (z_1+z_2)^*&=&z_1^*+z_2^*\\ (z_1\cdot z_2)^*&=&z_1^*\cdot z_2^*\\ (1)^*&=&1^* \end{array}$$
Proofs: 1
Propositions: 2
Theorems: 3
